World number two Murray can dethrone Novak Djokovic at the top of the rankings with a victory in the French capital
Verdasco, a former world number seven, was leading 6-2, 3-2 before a hip injury forced Haase to throw in the towel in their first-round encounter.
World number two Murray can dethrone Novak Djokovic at the top of the rankings with a victory in the French capital if the Serb fails to reach the final, or if the Briton makes the final and Djokovic falls before the semis.
Former Australian Open finalist Marcos Baghdatis defeated French wildcard Paul-Henri Mathieu 6-2, 6-4, while Benoit Paire joined the French exodus after losing 6-4, 7-6 (7/4) to Paolo Lorenzi.
Pablo Carreno Busta, who defeated Fabio Fognini to lift the Kremlin Cup in Moscow a week ago, dealt the Italian another loss in Paris, racing to a 6-3, 6-1 victory.
Djokovic takes on the winner of Tuesday's match between Nicolas Almagro and Gilles Muller after receiving a first-round bye, while US Open champion Stan Wawrinka begins against Germany's Jan-Lennard Struff, who thrashed Illya Marchenko 6-1, 6-1.
Tomas Berdych is ninth in the Race to London standings with the Czech needing to progress to the last four to remain in with a chance of playing at the end-of-season Tour finals next month.
He trails Dominic Thiem and Marin Cilic, a winner in Basel at the weekend, in the battle for the final two places in the eight-man field and must beat Portugal's Joao Sousa to remain in contention.